Pair correlations of Halton and Niederreiter Sequences are not Poissonian

Niederreiter and Halton sequences are two prominent classes of higher-dimensional sequences which are widely used in practice for numerical integration methods because of their excellent distribution qualities. In this paper we show that these sequences—even though they are uniformly distributed—fail to satisfy the stronger property of Poissonian pair correlations. This extends already established results for one-dimensional sequences and confirms a conjecture of Larcher and Stockinger who hypothesized that the Halton sequences are not Poissonian. The proofs rely on a general tool which identifies a specific regularity of a sequence to be sufficient for not having Poissonian pair correlations.

A computational investigation of the optimal Halton sequence in QMC applications

We propose the use of randomized (scrambled) quasirandom sequences for the purpose of providing practical error estimates for quasi-Monte Carlo (QMC) applications. One popular quasirandom sequence among practitioners is the Halton sequence. However, Halton subsequences have correlation problems in their highest dimensions, and so using this sequence for high-dimensional integrals dramatically affects the accuracy of QMC. Consequently, QMC studies have previously proposed several scrambling methods; however, to varying degrees, scrambled versions of Halton sequences still suffer from the correlation problem as manifested in two-dimensional projections. This paper proposes a modified Halton sequence (MHalton), created using a linear digital scrambling method, which finds the optimal multiplier for the Halton sequence in the linear scrambling space. In order to generate better uniformity of distributed sequences, we have chosen strong MHalton multipliers up to 360 dimensions. The proposed multipliers have been tested and proved to be stronger than several sets of multipliers used in other known scrambling methods. To compare the quality of our proposed scrambled MHalton sequences with others, we have performed several extensive computational tests that use {L_{2}} -discrepancy and high-dimensional integration tests. Moreover, we have tested MHalton sequences on Mortgage-backed security (MBS), which is one of the most widely used applications in finance. We have tested our proposed MHalton sequence numerically and empirically, and they show optimal results in QMC applications. These confirm the efficiency and safety of our proposed MHalton over scrambling sequences previously used in QMC applications.

Remarks on randomization of quasi-random numbers

In this paper we discuss estimation of the quasi-Monte Carlo methods error in the case of calculation of high-order integrals. Quasi-random Halton sequences are considered as a special case. Randomization of these sequences by the random shift method turns out to lead to well-known random quadrature formulas with one free node. Some new properties of such formulas are pointed out. The subject is illustrated by a number of numerical examples.